Some Observations on Dyson's New Symmetries of Partitions

We utilize Dyson’s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular” partitions with modulus 2. This way we find a new combinatorial proof of Gauss’s famous identity. We give a direct combinatorial proof that for n > 1 the partitions of n with crank k are equinumerous with partitions of n with crank −k.